Probability

A game is played by tossing two unbiased coins repeatedly until two heads are obtained in the same throw. The random variable X denotes the number of throws required. Find an expression for P(X=r)
Before playing the game, the player has to guess the value that X will take. If the player guesses correctly, he wins $5 . For an incorrect guess , the player loses $1. Suppose the player guesses X=t, express in terms of t, the expected amount won in a game.

I am not sure how to get the expression for part (1). If only one unbiased coin is tossed repeatedly, the probability of getting a head is

A game is played by tossing two unbiased coins repeatedly until two heads are obtained in the same throw. The random variable X denotes the number of throws required. Find an expression for P(X=r)
Before playing the game, the player has to guess the value that X will take. If the player guesses correctly, he wins $5 . For an incorrect guess , the player loses $1. Suppose the player guesses X=t, express in terms of t, the expected amount won in a game.

I am not sure how to get the expression for part (1). If only one unbiased coin is tossed repeatedly, the probability of getting a head is

If I understand to game correctly, the key concept here is the tossing of two coins and getting two heads.
The probability of that is .
So fhe probability of failure is .
Now set up the geometric distribution.

If I understand to game correctly, the key concept here is the tossing of two coins and getting two heads.
The probability of that is .
So fhe probability of failure is .
Now set up the geometric distribution.

Thanks Plato, but how many trials are there? It seems to be infinity?

X~B(? , 1/4)

but you seem to be suggesting that the probability is as such,

1/4+(1/4)(3/4)+(3/4)(3/4)(1/4)+...

=1/4(1+3/4+(3/4)^2+...)

but this is not an expression. Sorry if my answers and thoughts are totally off track and not of your expectation.